The present study theoretically explores combined dry and wet adhesive contact between a rigid sphere and elastic semi-half substrate, in which dry contact is encircled by liquid bridge. We consider threefold effects of liquid bridge on contact behavior, namely Laplace pressure induced by the curved surface of liquid meniscus, surface tension at the triple-phase junction and alternation of adhesion energy between solid surfaces ascribed to liquid immersion. A clear novelty in this study is the investigation on the effect of surface tension at the vapor-liquid-solid junction on the adhesive contact response, in contrast to previous studies. The model solution predicts that the contact behavior and adhesive strength are strongly dependent on surface wettability (manifested by contact angle), liquid volume and the contact system's rapidity in achieving thermodynamic equilibrium. It is found that the transition of the pull-off force is evidently different from Maugis-Dugdale model in terms of a couple of interesting characteristics. Moreover, it is unveiled that the jump instabilities and hysteresis of force-separation curves are highly affected by surface wettability and liquid volume. These theoretical results can not only shed lights on the mechanism of liquid-mediated adhesion employed by animals and plants, but also provide us inspiration for development of biomimetic adhesive devices.